Why does it sound better to say "the shiny little red car" than "the red little shiny car"?

Because of the royal order of adjectives.

I love that phrase. It makes it sound so ... regal. Or a secret club where you dress like those guards on bottles of Beefeaters gin (aka Yeomen Warders of Her Majesty's Royal Palace and Fortress the Tower of London, and Members of the Sovereign's Body Guard of the Yeoman Guard Extraordinary) and read the Oxford English Dictionary out loud for kicks.

The royal order of adjectives lays out the order in which categories of adjectives should fall before the nouns they modify.

Here's the not-so-secret sequence:

1. Determiner: articles (a, an, the), possessives (your, his, her, my, their, our), number, demonstratives (this, that, those, these)
2. Observation or opinion: anything that involves judgement or subjectivity (ugly, playful, nice, genuine, etc.)
3. Size: huge, tiny, petite
4. Shape: fat, thin, oblong
5. Age: old, young, ancient
6. Color: cerulean, sienna, ruby
7. Origin: American, Irish, Kenyan
8. Material: wood, silk, copper
9. Qualifier: an adjective integral to the noun (hound dog, racing car, wedding dress) or that describes its purpose (adding machine, walking stick, sewing machine)

Here's a ridiculous but technically correct example: "the gorgeous little old white American silk wedding dress." In other words, "determiner opinion size age color origin material qualifier noun." Mix up the order to hear how weird it would sound if you screwed up the royal order. 

As native English speakers, we know what sounds right and wrong. Most of us just didn't know it's because of the royal order of adjectives. Now you do.

By the way, notice the lack of commas between the adjectives? No commas are necessary between adjectives from different categories. But use commas to separate adjectives in like categories, such "an odd, ambivalent feeling" (observation/opinion) or "the long, cavernous blimp hangar" (size). When adjectives are equal, you can switch their order and still maintain proper syntax.

You are now an honorary member of the royal order of adjectives. Go forth and string your adjectives correctly.